The problem of broadcasting with noisy side information (BNSI) consists of a set of receivers, each demanding a subset of messages available at a sender. The receivers already have a noisy version of their demands (due to decoding errors in previous transmissions) that can be considered as noisy side information for the retransmission phase. The maximum possible number of errors in the noisy side-information at a receiver is referred to as its error threshold. The objective is to reduce the number of transmissions in the retransmission phase by taking advantage of the knowledge of noisy side-information at the receivers, when each receiver has a different error threshold. We consider linear encoding schemes and provide a set of necessary and sufficient conditions for any linear code to be valid. We present a field-size independent coding scheme with low-complexity (only additions allowed in the coding scheme). This generalizes a coding scheme given in a prior work for the BNSI problem with equal error thresholds. We then provide a greedy algorithm to obtain a field-size independent encoding scheme with low-complexity for any BNSI problem with equal error thresholds.

Groupcast index coding problem is the most general version of the classical index coding problem, where any receiver can demand any number of messages and have any subset of messages as side-information as described by its fitting matrix. A single-sender index coding problem is said to be a joint extension of a set of single unicast sub-problems, if the fitting matrices of all the sub-problems are disjoint submatrices of its fitting matrix. In our prior work (C. Arunachala and B. S. Rajan, "Optimal scalar linear index codes for three classes of two-sender unicast index coding problem", International Symposium on Information Theory and its Applications (ISITA 2018)), a special class of joint extensions were used to obtain scalar linear codes for some classes of two-sender unicast index coding problems. We extend the definition of single-sender joint extensions of single-unicast sub-problems to the case, where the sub-problems can also be groupcast problems. We study a special class of such joint extensions where the extended problem is also dependent on another groupcast problem called the base problem. The fitting matrix of the base problem decides the positions of fitting matrices of the sub-problems in that of the extended problem. We then provide an algorithm to construct a scalar linear code (not optimal in general), for this class of joint extensions using scalar linear codes of all the sub-problems and the base problem. We also identify a subclass where the constructed codes are scalar linear optimal.

The two-sender unicast index coding problem consists of two senders collectively having all the demanded messages of a set of receivers, where each receiver demands a unique message. The senders avail the knowledge of the side-information present at all the receivers to reduce the total number of broadcast transmissions. This problem is relevant in many practical communication problems like multi-source satellite communication, multi-user coded cooperative data exchange, and other related problems. In this paper, the two-sender unicast index coding problem is analyzed using three independent single-sender subproblems. Optimal broadcast rate (total number of transmitted bits per message bit as the message length tends to infinity) for all the unsolved instances of a special class of the two-sender unicast index coding problem is provided in terms of those of the three associated subproblems. The optimal broadcast rate established in this work serves as a lower bound for the optimal broadcast rate of any general associated two-sender unicast index coding problem. An achievable broadcast rate (total number of transmitted bits per message bit) with finite length messages for any finite length, is given for a subclass of the two-sender unicast index coding problem by providing a code-construction. This serves as a tighter upper bound when compared to the prior state of art.

The two-sender unicast index coding problem consists of two senders, each having a different set of messages. Some messages may be common to both the senders. Each receiver demands a unique message and has a subset of messages known as its side-information. The senders transmit coded messages by availing the knowledge of the side-information of all the receivers, such that all the receivers are able to decode their demands. The aim is to find the optimal aggregate number of coded transmissions per message length (also called the optimal broadcast rate with finite length messages), and its limiting value as the message length tends to infinity (also called the optimal broadcast rate). In this paper, only linear coding schemes are considered. Optimal linear broadcast rate for any finite message length and optimal linear broadcast rate for a basic class of the two-sender unicast index coding problem are established. Optimal code-constructions are also provided. These results are given in terms of the corresponding results of three independent single-sender sub-problems of the two-sender unicast index coding problem. Proof techniques used to obtain the results for the two-sender problem are shown to be useful in obtaining the results for some classes of the multi-sender unicast index coding problem.

The problem of two-sender unicast index coding consists of two senders and a set of receivers. Each receiver demands a unique message not demanded by any other receiver and has a subset of messages as its side information. Every demanded message is available with at least one of the senders. The senders avail the knowledge of the side information at all the receivers to reduce the total number of transmissions required to satisfy the demands of all the receivers. The objective is to find the minimum total of number of transmissions per message length (known as the optimal broadcast rate with finite length messages) and its limiting value as the message length tends to infinity (called the optimal broadcast rate). Achievable broadcast rates are provided for a class of the two-sender unicast index coding problem based on a special graph coloring technique called two-sender graph coloring. This result illustrates the utility of graph products in the two-sender unicast index coding problem for the first time in the literature. For another class, achievable broadcast rates are provided based on the optimal broadcast rates of three single-sender sub-problems with finite message length. This employs a code construction for the two-sender unicast index coding problem using optimal codes (including non-linear codes) of the sub-problems. Optimal broadcast rates are provided for a special class of the TUICP for which only an upper bound was known prior to this work. The optimal broadcast rates presented in this work also consider non-linear coding schemes at the two senders.

In this paper, two-sender unicast index coding problem (TUICP) is studied, where the senders possibly have some messages in common, and each receiver requests a unique message. It is analyzed using three independent subproblems (which are single-sender unicast index coding problems (SUICPs)) and the interactions among them. These sub-problems are described by three disjoint vertex-induced subgraphs of the side-information graph of the TUICP respectively, based on the availability of messages at the senders. The TUICP is classified based on the type of interactions among the sub-problems. Optimal scalar linear index codes for a class of TUICP are obtained using those of the sub-problems. For two classes, we identify a sub-class for which scalar linear codes are obtained using the notion of joint extensions of SUICPs. An SUICP IEis said to be a joint extension of l SUICPs if the fitting matrices of all the l SUICPs are disjoint submatrices of that of IE. Joint extensions generalize the notion of rank-invariant extensions. Scalar linear codes and a condition for optimality of the codes are given for a class of joint extensions. Using this result, scalar linear codes and the conditions for their optimality are obtained for two classes of the TUICP.

In bidirectional relaying using Physical Layer Network Coding (PLNC), it is generally assumed that users employ same modulation schemes in the Multiple Access phase. However, as observed by Zhang et al., it may not be desirable for the nodes to always use the same modulation schemes, particularly when node-relay channels are not equally strong. Such a scheme is called Heterogeneous PLNC. However, the above approach uses the computationally intensive Closest Neighbour Clustering algorithm to find the network coding maps to be applied at the relay. Also, the treatment is specific to certain cases of heterogeneous modulations. In this paper, we show that, when users employ heterogeneous symmetric-PSK modulations, the network coding maps and the mapping regions in the fade state plane can be obtained analytically. Performance results are provided in terms of relay error rate and bit error rate.